U n i v e r s i t y o f H e i d e l b e r g

Department of Economics

Discussion Paper Series No. 468

Too Few Cooks Spoil the Broth: Division of Labour and Directed Production

Marisa Ratto andWendelin Schnedler

July 2008

Too Few Cooks Spoil the Broth:Division of Labour

and Directed Production

Marisa RattoUniversite Paris-Dauphine (SDFi)

Wendelin Schnedler∗

Department of EconomicsUniversity of Heidelberg

First Version: September 2003This Version: July 2008

Abstract

How can a manager influence workers’ activity while knowing littleabout it? This paper examines a situation where production requiresseveral tasks, and the manager wants to direct production to achievea preferred allocation of effort across tasks. However, the effort thatis required for each task cannot be observed, and the production re-sult is the only indicator of worker activity. This paper illustratesthat in this situation, the manager cannot implement the preferredallocation with a single worker. On the other hand, the manager isable to implement the preferred allocation by inducing a game amongseveral workers. Gains to workers from collusion may be eliminatedby an ability-dependent, but potentially inefficient, task assignment.These findings provide a new explanation for the division of labor, andbureaucratic features such as ”over”-specialization and ”wrong” taskallocation.

Keywords: specialization, job design, moral hazard, multitasking

JEL-Codes: D02, D86, M54, D23, L23, J23

∗wendelin.schnedler@awi.uni-heidelberg.deThe authors wish to thank Guido Friebel, Paul Grout, Joop Hartog, Stefan Napel, JorgOechssler, Ulf Schiller, and Jan Zabojnik for their valuable comments. In addition, wethank participants of the Public Economic Theory conference in 2003 at Duke Univer-sity, at the Royal Economic Society annual meeting 2004, at the Kolloquium fur Per-sonalokonomie in Bonn 2004, at the Fifth Symposium of the German Economic Associ-ation for Business Administration in Bonn 2004, and at seminars in Bristol and Erfurt,who contributed to this paper.

Such schemes as these are nothing without numbers.One cannot have too large a party.

Jane Austen, Emma Vol. III, Chap. VI

1 Introduction

Division of labor has been a central theme in economic analysis since AdamSmith (1776) and a cornerstone of Frederick Winslow Taylor’s “scientificmanagement” (1911). Traditionally, the advantages of the division of la-bor have been attributed to the productive gains from learning by doing,economies of scale and comparative advantages. This paper argues that thedivision of labor has another advantage that has so far been neglected: ithelps to influence the production method, when only the production resultis verifiable.

In many production processes, little is known about how a particularresult has been achieved. For example, software may run well due to carefulengineering or due to extensive debugging; exam results can be improvedby better teaching or by rehearsing test situations; a placement officer maybe successful because she brushes up job seekers’ interview skills or becauseshe finds adequate vacancies. Although it is hard, or even impossible, todeduce how such outcomes are produced, it is often important to influencethe decision of how to produce because some production methods are moredesirable than others. Maintaining software is easier when it is well-designed;coping with test situations is only a secondary goal of the education system;job agencies prefer adequate and hence longer lasting placements.

This paper considers a multiple task principal-agent problem in whicheach task requires effort, the organizer of production (principal) prefers aspecific allocation of effort, agents who work for her prefer a different allo-cation, and the only evidence of production is the final result. The principalcannot enforce effort, but she can prevent agents from exerting effort as shecontrols necessary assets. For example, she can withhold tools and material,or prohibit entry to certain production areas. However, the power of theprincipal is rather limited, as all tasks are required, and she has to allow atleast one agent to use the task’s respective asset. Can the principal ensurethat production occurs in the desirable way despite such drastic limitations?

The paper shows that a single agent cannot be induced to produce in the

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desired way. The principal can pay the agent for a specific production result,and will even obtain this result, but the agent will focus effort according tohis preferences.

The central result of this paper is that division of labor allows the prin-cipal to implement her desired production method, even though individualor task-wise effort cannot be verified. The intuition is simple. Each agentcarries out only one task, and the principal pays on the basis of the overallresult. Thus, an agent can only affect this result—and hence his pay—bychanging his effort on his task. By adjusting the pay to a specific agent, theprincipal affects the effort of this agent and hence the effort exerted at thistask. As this can be done separately for every agent, the desired allocationcan be obtained. This result provides a novel explanation for the divisionof labor. Note that it is essential that the principal limits the agents’ dis-cretion over who carries out which task. Otherwise, agents will produce theresult while performing their preferred tasks, which may not correspond tothe desired production method.

A second finding is that the principal can implement her desired produc-tion method in a collusion-proof manner. Colluding agents have the samefreedom to obtain a result as a single agent. Similar to a single agent, col-luding agents can focus their effort on easy tasks, and reimburse the agentswho are working on these tasks. The principal, however, can make such de-viations unattractive by assigning easy tasks to unproductive agents. Thisfinding explains inefficient task assignments.

This paper is closely related to the work of Dewatripont and Tirole (1999),but differs at a crucial point. In the setting of Dewatripont and Tirole, anagent can obtain the same result (for example, a verdict) by slacking on twoconflicting tasks (for example, gathering less incriminating and exoneratingevidence). In many production contexts, the same result can only be ob-tained if the reduction of effort on one task is compensated by an increaseof effort on another task (for example, less careful programming is offset bymore debugging). This paper’s analysis covers such non-conflicting tasks.Specialization facilitates implementation, regardless of the nature of tasks.The implications of specialization are, however, very different. Where tasksare conflicting, incentive provision leads to competition. Competing agentsimpose a negative externality on each other. Where tasks are non-conflicting,agents have a joint interest in producing the result, and the externality is pos-itive. Secondly, competition among agents invites sabotage activities, whichare absent when tasks are non-conflicting. Finally, collusion is unavoidable

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with conflicting tasks, as agents gain by ceasing competition and slacking.In the situation analyzed here, collusion can sometimes be prevented by anappropriate assignment of agents to tasks.

The paper’s findings also extend and complement the literature on multi-task principal-agent models that were pioneered by Holmstrom and Milgrom(1991). Recent contributions to this literature analyze which informationabout the activity of a single agent needs to be verifiable in order to obtaina desired allocation of effort across tasks (Baker 2002, Schnedler 2006 andforthcoming). While these models tackle the problem that effort allocationcannot be identified from the production result,1 they do not consider thebeneficial effect of labor division.

Implementing the desired production method requires that several agentsbe remunerated on the basis of a joint production result.2 While the exis-tence of teams has been attributed to interaction in the production function(Alchian and Demsetz, 1972) or due to task-wise additive separable andconvex effort costs (Itoh, 1991), this paper contributes the alternative expla-nation that teamwork, rather than individual work, ensures a particular wayof production. All team members are indeed engaged in productive work,and this distinguishes this paper’s model from others in which incentivesare improved because agents supervise their colleagues (see Strausz 1997 orAthey and Roberts 2001). The remainder of the paper is organized as fol-lows. Section two sets up the model, and section three shows how the desiredproduction method can only be implemented if labor is divided. Section fourexamines when and how the desired production can still be implemented by aspecific task assignment when agents collude. Finally, section five concludes.

2 The Model

Consider a principal who wants to produce a good that involves two tasks.One task is easy (task E) and the other task is demanding (task D). Lackingthe time or skills to do the tasks herself, the principal employs one or two

1Similarly, talent and effort cannot be identified in career concern models (Fama 1980,Holmstrom 1999, Gibbons and Murphy 1992) and Dewatripont, Jewitt, and Tirole (1999)show that rotating tasks can solve this identification problem.

2Of course, joint production has also costs. Notably, workers may free-ride on theircolleagues’ effort. In our analysis these costs do not feature because the principal acts asa budget breaker (Holmstrom, 1982).

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agents to do the job. Each task involves an effort choice by agent i whocarries out the task: eD

i and eEi , where both are non-negative real numbers

from a bounded interval. Agents are allowed to mix over pure effort choices,and the corresponding cumulative distribution function for agent i is denotedby Fi while Pi(ei = e) is the probability that i chooses e if e is a mass point.

In line with the examples from the introduction, the principal cares abouthow production is achieved. Denote the effort choices desired by the principalby eD∗ for the difficult task and eE∗ for the easy task. In the following,we examine the conditions under which these desired effort choices can beimplemented.

The model assumes that effort is costly for the agent. In order to elimi-nate economies of scale and other incentives to specialize, the model assumesthat costs are additively separable. The cost of agent i in task k, cki , isa continuously differentiable function of effort that increases and is strictlyconvex. 3 The effort levels that an agent is willing to exert without incen-tives are denoted by eD0

and eE0, and the respective costs are normalized to

zero: cDi (eD0) = cEi (eE0

) = 0. In order to make the implementation probleminteresting, the model assumes that eD0 6= eD∗ and eD0 6= eD∗

We attach meaning to the labels “difficult” and “easy” of the two tasksby assuming that the marginal costs of both tasks are not identical given thedesired allocation (eD∗ , eE∗) and associating the difficult task with highermarginal costs:

dcDide

∣∣∣∣eD=eD∗

>dcEide

∣∣∣∣eE=eE∗. (1)

If the production method (the effort choices) could be verified by a courtof law, the principal could hire a single agent and stipulate the desired effortlevels in a contract. This contract would then work as a device to direct theagent’s effort. In many settings, however, the only evidence is the result ofproduction R, and it is not always clear how this result has been achieved.Accordingly, the model assumes that contracts about task-wise efforts cannotbe written, and that anything that reveals how the result has been generated(for example, intermediate stages of production) cannot be verified.4 In

3More generally, the agent may not exert effort but have a choice which affects himselfas well as the principal. However, ordering these choices according to costs and speakingof effort greatly simplifies the illustration.

4It may well be possible that they can be verified later when the agent can no longerbe held responsible—perhaps because he is working for another employer or retired.

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short, the benefit of the principal is private information, and the action ofthe agent is hidden. The contractual environment is thus that of a ”copingorganization” according to Wilson (1989). Summarizing these considerations,the model assumes that the only verifiable quantity R is a function of bothefforts. Notice that R is not stochastic. Still, R is imperfect as it confoundseffort on both tasks and does not reveal exactly what the agent did.5

The principal may care about the production result R; for example, ifshe sells the product on a market. More importantly, she also cares abouthow the result is achieved. For example, neglecting the demanding taskmay lead to a higher probability of product breakdown, and damage thefirm’s reputation, or it may lead to more wear- and-tear of the productionequipment, and necessitate costly repairs.6

In their seminal paper on advocates, Dewatripont and Tirole (1999) alsosuppose that the only verifiable variable, whether a culprit is convicted ornot, is the result of two effort choices: searching for incriminating evidence,and searching for exculpatory evidence. These two efforts are conflicting. Alarger R, say, conviction, can be obtained by increasing effort on one task, forexample, searching harder for incriminating evidence, but also by decreasingeffort on another task, for example, looking less hard for exculpatory evi-dence. In many production settings, it is not possible to improve a result byreducing effort. To the contrary, effort needs to be increased at least in onetask to obtain a larger outcome. In these settings, tasks are non-conflicting,because the agent does not nullify the effect of effort in one task by increas-ing effort in another task. The paper examines such production settings, andassumes that the result R is an increasing (concave and twice continuouslydifferentiable) function of effort exerted in the two tasks.

The result from the desired effort choice is denoted by r∗ := R(eD∗ , eE∗).An immediate consequence of the fact that the outcome increases and iscontinuous in both efforts is that the outcome r∗ can also be achieved bysome other effort choice. The crucial assumption of our model is that giventhe desired allocation (eD∗ , eE∗), increasing effort on the easy task is at leastas effective in generating a larger outcome than increasing effort on the de-

5The crucial mechanism that drives our results works also if the output measurementis noisy. However, eliminating noise simplifies the analysis.

6See the introduction for additional examples in which not only output but also themanner of production matters.

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manding task:

∂

∂eER(eD, eE)

∣∣∣∣eD=eD∗ ,eE=eE∗

≥ ∂

∂eDR(eD, eE)

∣∣∣∣eD=eD∗ ,eE=eE∗. (2)

This assumption reflects the idea that often, there is an easier way thanthe desired one to produce an output. This paper will show that the presenceof this opportunity may render the implementation of the desired productionmethod impossible.

So far, the model has assumed that the production outcome is the onlyverifiable result of the agent’s effort choice. By conditioning transfers τ onthis outcome, the principal can influence the agent’s choice. The model’s finalassumption is that the principal has another more basic, but very limited,method to control what the agent is doing. She can assign tasks to one ormore agents. More specifically, she can prevent agents from carrying out atask. In practice, this could be done by withholding material or instrumentsneeded for the task, restricting access to a location at which the task isperformed, or refraining from training the worker to carry out the task. Themost common form of prevention is prohibiting the agent from carrying outa task. In many circumstances, it is very simple to describe what the agentshould not do, and much harder to describe what the agent should do. Forexample, there is, to our knowledge, no verifiable definition of good teachingpractice, while it is relatively simple to spot whether a class is rehearsing atest. Whenever the principal controls an asset required for a task, she canprevent the agent from carrying out this task. If, however, the principalprovides the necessary asset, this by no means implies that the agent will usethe asset in the desired way.

3 Directing production

This section examines the organizational form within which the principal canimplement the desired production method. The section first looks into thesingle-agent case, and then moves to the two-agent case.

3.1 Single-agent case

In the case where the principal only employs one agent, implementation isonly possible if the agent is allowed to carry out all tasks. The implementa-

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tion problem then consists in finding transfers τ at which the agent is willingto exert the desired effort allocation.

In order to implement the desired effort allocation, the agent has to berewarded whenever the production result looks like it has been achieved bysuch allocation. In this case, however, he will produce this result in thecheapest, and not necessarily the most desired, way. Formally, the followingresult holds:

Proposition 1 (Window-dressing by a single agent). If there is a single agenti and the principal provides incentives to implement r∗, then the agent canprofitably engage in window dressing, i.e. reduce effort on the demanding taskand increase effort on the easy task. The desired production method cannotbe implemented.

Proof. To implement the desired production method, it is a necessary con-dition that the agent creates the result r∗. Whatever the incentive schemeused to implement (eD∗ , eE∗) and hence r∗, the agent can obtain the sameresult and transfer by increasing effort on the easy task and reducing efforton the demanding task. The marginal reduction can be computed using the

implicit function theorem: ddeE e

D = −∂

∂eE R(eD,eE)∂

∂eD R(eD,eE). This derivative is larger

than one for (eD∗ , eE∗) by Equation (2), thus increasing eE by one unit al-lows the agent to reduce eD by at least one unit. The loss from increasingeffort in the easy task is outweighed by the gain from reducing effort in thedifficult task by Equation (1). So, deviating from (eD∗ , eE∗) is profitable forthe agent.

The main message of this proposition is that any attempt by the princi-pal to achieve the desired production method is doomed, because the agentalways prefers a different effort allocation (refer to Figure 1). More specif-ically, the agent always slacks on the demanding task, and brushes up theappearance using the easy task. The crucial feature of the verifiable outcomeR, which drives this result, is that it confounds the efforts of two (non-conflicting) tasks. This feature, together with the (local) preference for theeasy task, suffices for the agent to engage in window-dressing rather thancarrying out production in the desired way. This problem is not specific todeterministic environments. As long as the task on which the agent focuseseffort matters, and as long as this effort cannot be identified by looking atthe production result, window-dressing is possible.

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The principal can circumvent this problem by assigning the easy task toanother agent. This will be explored in the following sub-section.

Figure 1: Deviating from the desired production method by slacking on thedemanding task (−∆eD) and working harder on the easy task (∆eE) leads tothe same observable result (window dressing), while the agent gains (2)-(1).

eD*e

E*e eE

easy effort

demanding effort

demandingtask

costs of

easytask

costs of

D

agent with low marginal costs

easy effort

demanding effort

eD*e D

E*e e E

agent with high marginal costs

demandingtask

costs of

easytask

costs of

cost

sco

sts

2

1

3

3.2 Two-agent case

In this section, we analyze the situation in which the principal hires twoagents, and assigns them to the two tasks. This type of specialization enablesthe principal to implement the desired production method.

Proposition 2 (Implementation of the desired production method). If agentsare assigned to different tasks (specialization), transfers can be chosen suchthat the desired production method (eD∗ , eE∗) can be implemented as a (unique)Nash equilibrium.

Proof. In order to show that (eD∗ , eE∗) can be implemented as a (unique)Nash equilibrium, we consider two agents i = 1, 2, specify transfers, elimi-nate some strictly dominated strategies and compute the best-reply corre-spondences. For notational convenience, we focus on agent 1 and let himwork on the demanding task (the analysis for agent 2 is completely analo-

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gous). Define the following transfer scheme:

τ1 :=

cD1 (eD∗) if R(e1, e2) = r∗

−K if R(e1, e2) = R(eD0, eE0

) =: r0

0 else,

where K > 0. For the transfers to agent 2 replace eD∗ by eE∗ .Before determining best-reply correspondences given these transfers, we

show that efforts that lead to negative payoffs are strictly dominated. Ob-serve that agent 1 can always obtain a payoff of at least −δ, where δ is anarbitrary small positive number, irrespective of the action chosen by agent 2.The respective strategy is to exert slightly more effort than eD0

, say eD0+ ε.

The same holds for agent 2. Since effort above the desired level eD∗ alwaysleads to a negative payoff, these efforts are strictly dominated. In equilibriumthe support of the strategy of agent 2 is hence a subset of E2 := {e2 ≤ eE∗}.

Given this observation about E2, the payoff to agent 1 from a (possiblymixed) strategy F1 given F2 amounts to:∫ ∫

(τ1(R(e1, e2))− cD1 (e))dF1(e1)dF2(e2) =∫E2\{eE∗ ,eE0}

∫(τ1(R(e1, e2))− c1(e1))dF1(e1)dF2(e2) (3)

+ P2(e2 = eE∗) ·∫

(τ1(R(e1, eE∗)− c1(e1))dF1(e1) (4)

+ P2(e2 = eE0

) ·∫

(τ1(R(e1, eE0

)− c1(e1))dF1(e1). (5)

Note that this payoff can at most be zero: the integrand of the integral inline (3) is zero only if e1 = eD0

and otherwise negative; the integrand of theintegral in line (4) is zero only if e = eD∗ or e = eD0

and otherwise negative;the integrand in line (5) is always negative.

We use this payoff to compute the best-replies. We decompose the strat-egy space of agent 2 into three cases.Case 1: P2(e2 = eE0

) > 0. Agent 1’s payoff is negative due to line (5) andhe can profitably deviate to eD0

+ ε with ε chosen to be sufficiently small.Therefore, there is no best-reply for agent 1 in this case.Case 2: P2(e2 = eE0

) = 0 and P2(e2 = eE∗) < 1. Line (3) implies thatany strategy with P1(e1 = eD0

) < 1 yields negative payoff and is strictlydominated by eD0

. Thus, the best-reply in this case is eD0.

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Case 3: P2(e2 = eE0) = 0 and P2(e2 = eE∗) = 1. Only line (4) matters, and

any strategy with P1(e1 = eD0) + P1(e1 = eD∗) < 1 yields a negative payoff.

Thus, the support of the best-reply is eD0and eD∗ in this case.

A completely analogous argument for agent 2 shows that (eD∗ , eE∗) is theonly fixed-point of the best-reply correspondences.

The proof of this proposition is constructive and provides specific transfersthat implement the desired production method. These transfers have tofulfill two conditions: (i) they need to work for the general class of resultfunctions considered, and (ii) they need to implement the desired methodas a unique Nash equilibrium. Both conditions are met by discontinuoustransfers that reward the desired outcome r∗, and punish the outcome fromlow efforts r0. The discontinuity implies that there is no best-reply if theother agent exerts the minimal effort, and transfers are reminiscent of ascheme used by Holmstrom (1982) in order to approximate the first-bestsolution in a single-task setting with noise (Theorem 3 in Holmstrom, 1982).Differing from Holmstrom’s scheme, where the approximation requires everlarger punishments that occur with ever lower probabilities, the punishmenthere can be arbitrarily small. Moreover, the discontinuity is not essentialfor implementation as such (as in Holmstrom’s argument), but only ensuresuniqueness (see appendix). For specific result functions, it is possible toimplement the desired allocation as a unique equilibrium even if transfersare continuous (for example, if the marginal effect of an agent on the resultincreases in the effort of the other agent—see appendix). The key ingredientin any mechanism to implement the desired allocation is that the principalgains an extra degree of freedom by having two agents: task separation allowsher to set incentives separately for the two tasks.

The implementation of the desired production requires that the agentwho is responsible for the demanding task has no access to the easy task.Otherwise, this agent could again profitably engage in window dressing.

The fact that the agent who carries out the demanding task has to beprevented from carrying out the easy task can entail direct costs, such as notbeing able to access certain areas or use certain tools. These costs may renderit more difficult for him to carry out his tasks. Similarly, there may be a pricefor installing the necessary technology to ensure that the agent assigned tothe demanding task is not working on the easy task. Moreover, hiring theadditional agent can be costly, and finding an equally productive agent maynot be possible. Proposition 2 provides a reason why specialization occurs

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despite all these costs, and in the absence of the advantages that are typicallyassociated with specialization. Thus, it explains why a principal hires a lessproductive or otherwise costly agent, although the same observable resultcould be produced less expensively by a single agent.

The desired production method is achieved while the production technol-ogy remains the same. Moreover, the only contractual variable, the resultof production,R, is also exactly the same. The crucial difference is that theprincipal employs two agents instead of one. Accordingly, the desired pro-duction may no longer be possible if the two agents act as if they were oneagent. In other words, the two agents might collude, and thereby underminethe incentive scheme. This problem is explored in the following section.

4 Collusion

Collusion is only possible when agents have a larger contract space than theprincipal, so as to formally or informally contract on the easy task. For exam-ple, agents may engage in repeated interactions while the principal is beingreplaced.7 We define the implementation of an effort choice e as collusion-proof when there is no contract among agents that stipulates transfers andeffort choices different from e such that at least one agent is better off andnone of them is worse off.8

If two agents collude, they will minimize their joint costs. If their marginalcosts are identical, they are in the same position as a single agent. In this case,division of labor cannot help to implement the desired production method.However, the fact that all agents find one task demanding and the otherone easy does not mean that all agents have to have identical marginal costs.There may, for example, be strong agents whose marginal costs for both tasksare lower than those of weak agents. This difference in marginal costs enablesthe implementation of the desired production method even when agents cancollude. If a strong agent is assigned to the demanding task and a weakagent to the easy task, the gains from slacking on the demanding task maybe more than offset by the loss from greater effort on the easy task. Figure 2provides an example.

7Roy (1952) reports on workers’ behavior in a production line where workers were ableto enforce effort levels in order to restrict joint output, while the employer was unable toobserve it.

8This definition of collusion-proofness is in line with Tirole (1986).

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Figure 2: The gains to an agent with low marginal costs (2) from slackingare offset by the costs to the agent with high marginal costs (3) from highereffort, and collusion does not pay.

eD*e

E*e eE

easy effort

demanding effort

demandingtask

costs of

easytask

costs of

D

agent with low marginal costs

easy effort

demanding effort

eD*e D

E*e e E

agent with high marginal costs

demandingtask

costs of

easytask

costs of

cost

sco

sts

2

1

3

12

Proposition 3 (Collusion-proof implementation). A collusion-proof imple-mentation of the desired production method is possible if and only if thereis a (specialized) assignment of agents to tasks such that the desired way ofproduction minimizes the costs of producing r∗, i.e.,

(eD∗ , eE∗) ∈ argmineD,eE{cDi (eD) + cEj (eE)|R(eD, eE) = r∗},

where i refers to the agent on the difficult task and j to that on the easy taski 6= j.

Proof. As a preliminary step, let us examine the problem of minimizing thecosts of producing r∗:

mineDi ,eE

j

cDi (eD) + cEj (eE) such that R(eDi , e

Ej ) = r∗. (6)

Since cDi + cEj is a continuous function, efforts are bounded and the side-constraint defines a closed set, the minimizer always exists. Let us denotethe minimizer by e.

The first step of the proof is to illustrate that it is impossible to im-plement e∗ := (eD∗ , eE∗) in a collusion-proof manner when this choice doesnot minimize the costs of producing r∗: (eD∗ , eE∗) 6∈ argmineD,eE{cDi (eD) +cEj (eE)|R(eD, eE) = r∗}. Suppose a respective mechanism exists, then theagents can write a contract that stipulates e and leads to the same observ-able result r∗. The agent with a higher effort under this contract can becompensated for the additional effort by the other agent, and the latter isstill strictly better off because the joint costs of production are lower for ethan for e∗. Accordingly, the implementation is not collusion-proof.

The second step of the proof is to show that it is possible to implement e∗

in a collusion-proof manner when it minimizes production costs (eD∗ , eE∗) ∈argmineD,eE{cDi (eD)+cEj (eE)|R(eD, eE) = r∗}. For this step of the proof, takethe transfers from Proposition 2. The payoff from (eE∗ , eD∗) is non-negative,whereas any result different from r∗ does not yield a payoff that is larger thanzero. So, agents cannot gain from a contract that stipulates efforts that leadto a result different from r∗. Within the effort allocations that lead to r∗, thedesired production method has the lowest costs. Thus, there is no contractthat leads to a weakly better payoff for both agents and to a strictly betterpayoff for at least one agent. The implementation is collusion-proof.

Again, the ability of the principal to prevent agents from carrying outcertain tasks is crucial here. If the strong agent has access to the easy task,

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agents would maximize their joint surplus by letting the strong agent doall the work. Then, window dressing by this agent would be a profitabledeviation from the desired way of production.

It would be wrong to conclude from this proposition that the negativeconsequences of collusion can generally be avoided. This requires a specificrelationship between the costs of agents and it may be difficult to find suchagents. The key idea that is formalized by this proposition is that deviationsfrom the desired allocation can be made more costly by assigning an agentwith high marginal costs to the easy task, and an agent with low marginalcosts to the demanding task. This idea allows the principal to obtain anallocation that is closer to the desired one, even if the desired allocationitself cannot be achieved.

Notice that the assignment of agents to tasks is entirely driven by thedesire to implement a certain allocation. This assignment is based on thestrong agent’s marginal costs in the demanding task and the weak agent’smarginal costs in the easy task. In particular, the assignment is independentof the relative marginal costs of agents with respect to tasks. Thus, agentsmay not be assigned to tasks where they have a comparative advantage. Inother words, an inefficient task assignment may be the only way to ensurethat the desired allocation (or an allocation close to it) can be implementedin a collusion-proof manner. This could explain why large organizations andbureaucracies that are designed to achieve a specific goal are often plaguedby inefficient allocation of workers to tasks.

5 Conclusion

Consider a scenario in which the only evidence of an agent’s performanceis the final result of production, and there is no way of verifying how suchresult has been achieved. This paper has shown that in this scenario, it isnot possible to direct a single agent’s effort toward tasks in a desired way.However, this problem can be overcome by splitting the production processsuch that agents are assigned specific tasks, and are prevented from carryingout the tasks that other agents are assigned to.

Division of labor entails costs that should be traded off against its bene-fits. These costs include the costs of hiring workers, the costs of communica-tion and coordination among workers (refer to Becker and Murphy 1992 orBolton and Dewatripont 1994), the forgone benefits from task complemen-

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tarities (Lindbeck and Snower, 2000), and the possibly higher compensationrequired by workers that like variation (i.e. workers with convex costs thatare additively separable across tasks, see Itoh 1992). The costs of labordivision increase further if such division has to be enforced.

The first main insight of this paper is that there are additional benefits tothe division of labor that have so far been neglected, which explain why divi-sion of labor may occur where it is not expected (“over”-specialization). Theexistence of these benefits is relevant to the trade-off between a tayloristicand holistic organization of production. Lindbeck and Snower (2000) arguethat new versatile technologies which make workplaces more flexible, such ascomputers, diminish the gains from specialization. Accordingly, productionunder new technologies should be less divided. However, this paper showsthat if the versatility of the new technology renders it more difficult to deter-mine how a production result has been achieved, then the diminishing gainsfrom specialization may be offset by the incentive gains from the division oflabor

The second main insight of this paper is that inducing a game betweenagents helps implement activities, even when agents are not played off againsteach other as in Dewatripont and Tirole’s article on advocates (1999). Thepaper also explains how it is possible to prevent collusion among agents toundermine the implementation in this case. This requires, however, thatagents’ productivities differ in a specific way, and that the less productiveagent be assigned to the easy task. For this assignment, the principal hasto know the costs of the agents. If these costs are private information, theprincipal should implement a mechanism to elicit them. Our conjecture isthat such a mechanism may be constructed by allowing agents to self-selecttasks, and giving the agent assigned to the easy task the right to denouncean unproductive colleague assigned to the demanding task. Designing sucha mechanism, however, is beyond the scope of this paper, and it is left tofuture research. Interestingly, the assignment of agents to tasks in a waythat prevents collusion does not necessarily coincide with the comparativeadvantages of the respective agents. This may explain the apparent absurdityof situations where agents are prevented from carrying out a simple task andhave to rely on a less able co-worker, although it would be cheaper if theywere in charge of the whole production.

Both features, “over”-specialization and inefficient job assignment, areoften associated with bureaucracies. Prendergast (2003) points out that bu-reaucracies may be regarded as an optimal solution to the problem of pro-

15

viding incentives, when important quantities are not tangible and cannot becontracted upon. This paper extends Prendergast’s observation to hiring de-cisions and work assignments in bureaucracies: seemingly inefficient hiringsand assignments can ensure the optimal provision of incentives in a sparsecontractual environment.

References

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Athey, Susan and John Roberts. 2001. Organizational design: Decision rightsand incentive contracts. American Economic Review 91, no. 2:200–205.

Baker, George. 2002. Distortion and risk in optimal incentive contracts.Journal of Human Resources 37, no. 4:728–751. Special Issue on DesigningIncentives to Promote Human Capital.

Becker, Gary S. and Kevin M. Murphy. 1992. The division of labor, coordina-tion costs, and knowledge. Quarterly Journal of Economics 107:1137–1160.

Bolton, Patrick and Mathias Dewatripont. 1994. The firm as a communica-tion device. Quarterly Journal of Economics 109, no. 4:809–839.

Dewatripont, Mathias, Ian Jewitt, and Jean Tirole. 1999. The economicsof career concerns, part II: Application to missions and accountability ofgovernment agencies. Review of Economic Studies 66:199–217.

Dewatripont, Mathias and Jean Tirole. 1999. Advocates. Journal of PoliticalEconomy 107, no. 1:1–39.

Fama, Eugene F. 1980. Agency problems and the theory of the firm. Journalof Political Economy 88, no. 2:288–307.

Gibbons, Robert and Kevin J. Murphy. 1992. Optimal incentive contracts inthe presence of career concerns theory and evidence. Journal of PoliticalEconomy 100, no. 3:469–505.

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Holmstrom, Bengt. 1982. Moral hazard in teams. Bell Journal of Economics13, no. 2:324–340.

Holmstrom, Bengt. 1999. Manegerial incentive problems: A dynamic per-spective. Review of Economic Studies 66:169–182.

Holmstrom, Bengt and Paul Milgrom. 1991. Multitask principal-agent-analysis: Incentive contracts, asset ownership, and job design. Journalof Law, Economics, and Organization 7:24–52. Special issue.

Itoh, Hideshi. 1991. Incentives to help in multi-agent situations. Economet-rica 59, no. 3:611–636.

Itoh, Hideshi. 1992. Cooperation in hierarchical organizations: An incentiveperspective. Journal of Law, Economics and Organization 8, no. 2:321–345.

Lindbeck, Assar and Dennis Snower. 2000. Multitask learning and the reor-ganization of work: From tayloristic to holistic organization. Journal ofLabor Economics 18, no. 3:353–376.

Prendergast, Canice. 2003. The limits of bureaucratic efficiency. Journal ofPolitical Economy 111, no. 5:929–958.

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Tirole, Jean. 1986. Hierarchies and bureaucracies: On the role of collusion inlarge organisations. Journal of Law, Economics, and Organization 2:181–214.

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Appendix

Proposition 4 (Implementation with continuous transfers). There is a con-tinuous transfer scheme τ such that the desired production (eD

1 = eD∗ , eE2 =

eE∗) can be implemented as a Nash equilibrium.

Proof. Denote by k(i) the task carried out by agent i. Consider the linear

transfers: τi(r) =c′i(e

k(i)∗ )∂R(eD,eE)

∂ek(i)|(eD,eE)=(eD∗ ,eE∗ )

· r. Observe that the payoff of agent

i is concave and has a unique maximizer for any e−i. This best-reply is char-

acterized by the first order condition:c′i(e

k(i)∗ )∂R(eD,eE)

∂ek(i)|(eD,eE)=(eD∗ ,eE∗ )

∂R∂ek(i) |(eD,eE) −

c′i(ek(i)) = 0. For e−i = ek(−i)∗ , the maximizer is hence ek(i)∗ .

Proposition 5 (Unique implementation with continuous transfers). If

∂∂R(eD, eE)

∂eD∂eE≥ 0 for all eD and eE,

there is a continuous transfer scheme τ such that the desired production(eD∗ , eE∗) can be implemented as a (unique) Nash equilibrium.

Proof. For notational simplicity, let agent i work on the demanding task (thedefinitions and analysis for the other agent are completely analogous). Definee−i(r) implicitly via R(eD∗ , e−i) = r. Consider the following transfers τi suchthat τi(r) is constant for r ≥ r∗ and twice continuously differentiable forvalues r < r∗ with the derivative:

τ ′i(r) =c′i(e

D∗)∂R(eD,eE)

∂eD |(eD,eE)=(eD∗ ,e−i(r))

,

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where the equality also holds for the limes r → r∗ with r < r∗. The secondderivative of the transfer in the outcome is:

τ ′′i (r) = − c′i(eD∗)

(∂R(eD,eE)∂eD |(eD,eE)=(eD∗ ,e−i(r)))

2·∂∂R(eD, eE)

∂eD∂eE

∣∣∣∣(eD,eE)=(eD∗ ,e−i(r))

·∂e−i(r)

∂r.

This second derivative is not positive because ∂∂R(eD,eE)∂eD∂eE ≥ 0 and ∂e−i(r)

∂r> 0.

Then, consider the payoff to agent i given a pure strategy e−i: τi(R(ei, ej))−ci(ei). Since τi is a concave function in R and R itself is concave in effortei, the first term is concave. Together with the observation that ci is strictlyconvex, we obtain that the payoff given a pure strategy is strictly concave. In-tegrating this strictly concave function over the possible values for e−i yieldsalso a concave function and thus there is a unique best-reply. Using con-tinuous differentiability of the integrand and the intermediate value theoremfor integrals, the payoff given any mixed strategy of the other agent is equalto some pure strategy e−i. Without loss of generality, focus is restricted tothese strategies. First, consider the best-reply to e−i ≤ eE∗ . By construction,the derivative of the agent i’s payoff is zero at eD∗ . Concavity implies thatthe first-order condition describes a maximum. The best-reply is hence eD∗ .Second, consider the best-reply to e−i > eE∗ . Then, the maximal transfer,which occurs at r∗, can be achieved with an effort ei < eD∗ . So, the bestreply is below eD∗ .

If k(i) denotes the task carried out by agent i, the best reply is ek(i)∗ fore−i ≤ ek(−i)∗ and below ek(i)∗ for e−i > ek(−i)∗ . So, the only fixed point isat (eD∗ , eE∗). Since the best-reply to any strategy is a unique pure strategy,there are no mixed-strategy equilibria either.

The condition, ∂∂R(eD,eE)∂eD∂eE ≥ 0, establishes that the transfers suggested

in the proof are concave in the production result. As a result, the agent’spayoff is concave and the agent has a unique and pure best-reply. However,transfers do not have to be concave for the agent’s payoff to be concave.Thus, violations of this condition are possible, and the desired efforts canstill be implemented using the same continuous scheme.

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